A new upper bound for the chromatic number of a graph

Let G be a graph of order n with clique number !(G); chromatic number ´(G) and independence number fi(G): We show that ´(G) • n+!+1ifi 2 : Moreover, ´(G) • n+!ifi 2 ; if either ! + fi = n + 1 and G is not a split graph or fi+! = ni1 and G contains no induced K!+3iC5:

[1]  L. Lovász.  Three short proofs in graph theory , 1975 .

[2]  Ronald D. Dutton,et al.  A compilation of relations between graph invariants , 1985, Networks.

[3]  E. A. Nordhaus,et al.  On Complementary Graphs , 1956 .

[4]  Ingo Schiermeyer,et al.  Vertex Colouring and Forbidden Subgraphs – A Survey , 2004, Graphs Comb..

[5]  S. E. Markosyan,et al.  ω-Perfect graphs , 1990 .

[6]  Paul D. Seymour,et al.  Recognizing Berge Graphs , 2005, Comb..

[7]  Robin Thomas,et al.  Progress on perfect graphs , 2003, Math. Program..

[8]  Jonathan L. Gross,et al.  Handbook of graph theory , 2007, Discrete mathematics and its applications.

[9]  R. L. Brooks.  On colouring the nodes of a network , 1941, Mathematical Proceedings of the Cambridge Philosophical Society.

[10]  P. Erdös,et al.  Graph Theory and Probability , 1959 .